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          <h1 class="post-title" itemprop="name headline">卡方分布及卡方检验</h1>
        

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        <p><a href="https://blog.csdn.net/cuitzjd/article/details/80755310" target="_blank" rel="noopener">Doc1</a><br><a href="https://cosx.org/2010/11/hypotheses-testing/" target="_blank" rel="noopener">Doc2</a></p>
<h3 id="统计检验"><a href="#统计检验" class="headerlink" title="统计检验"></a>统计检验</h3><ul>
<li>出发点：小概率事件发生则假设不成立。可以检验一个样本与总体的差异，也可以检验一个样本集与另一个样本集的差异。</li>
<li>e.g. 观察到某个班级男女生身高的均值有差距，要证明这一点，先假设其”没有差距“，然后做相关的检验，如果检验出来有差距这个事件的概率小于某个值，则说明假设不成立。即一个具有足够小概率的事件发生了，这不是偶然。</li>
<li>第一类错误：原假设为真，检验的结果劝你放弃原假设。其概率为α。显著性水平。</li>
<li>第二类错误：原假设为假，检验的结果劝你接受原假设。概率为1-α。</li>
<li>显著性检验，即只限定第一类错误。当经过检验发现p&gt;α，则说明检验、总体这两样本之间不存在显著性差异，因此接受原假设；否则，说明一个很小概率的事件发生了，不接受原假设。</li>
<li>不同的检验方式有不同的前提。</li>
</ul>
<h3 id="T"><a href="#T" class="headerlink" title="T"></a>T</h3><ul>
<li><p><strong>用于样本量较小，总体标准差σ未知的正态分布。</strong></p>
</li>
<li><p>单总体检验：一个样本的平均数与已知的总体平均数的差异是否显著。</p>
</li>
<li><p>双总体样本检验：一个样本与另一个样本的差异。</p>
</li>
</ul>
<h3 id="F"><a href="#F" class="headerlink" title="F"></a>F</h3><ul>
<li>两个样本的总体方差是否相等。</li>
</ul>
<h3 id="卡方"><a href="#卡方" class="headerlink" title="卡方"></a>卡方</h3><h4 id="卡方检验"><a href="#卡方检验" class="headerlink" title="卡方检验"></a>卡方检验</h4><p><img src="https://ss0.bdstatic.com/94oJfD_bAAcT8t7mm9GUKT-xh_/timg?image&amp;quality=100&amp;size=b4000_4000&amp;sec=1512961717&amp;di=6133a04e10dc72b22bfb76fca025156d&amp;src=http://pic.baike.soso.com/p/20130619/20130619123748-1653885327.jpg" alt></p>
<ul>
<li><p>原假设为两者分布无差异、不相关等。</p>
</li>
<li><p>卡方值越大，实际（假设）与理论（根据原假设求得的概率值/统计量）差异越大，即假设的错误程度越大，超过阈值，则推翻假设。</p>
<p>若求得的卡方值落在阈值之上，说明小概率事件(e.g. 5%)发生了，则推翻原假设，说明两者差异的程度确实够大，即两者有差异；否则有95%的置信程度说明两者无差异。</p>
</li>
<li><p>参数有：自由度、置信水平。</p>
<p>例如，看喝牛奶与感冒有没有关系。给定置信度β，确定卡方值阈值x。</p>
<p>原假设：无关。</p>
<p>卡方值：先在原假设的基础上求出理论分布，然后看理论分布与实际分布的差异性（卡方值）。</p>
<p>查表，看有没有超过阈值，若超过，说明小概率（显著性水平：1-β）事件发生，推翻原假设，两者有关。</p>
</li>
</ul>
<h4 id="卡方分布"><a href="#卡方分布" class="headerlink" title="卡方分布"></a>卡方分布</h4><p>n个相互独立且均服从标准正态分布的随机变量，其平方和服从卡方分布。</p>
<p><img src="https://img-blog.csdn.net/20180109180130184?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvc25vd2Ryb3B0dWxpcA==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt></p>
<h4 id="卡方分箱"><a href="#卡方分箱" class="headerlink" title="卡方分箱"></a>卡方分箱</h4><ul>
<li><p>算法步骤：</p>
<ul>
<li><p>初始化：排序，每个实例属于一个一个区间</p>
</li>
<li><p>合并区间：</p>
<ul>
<li><p>计算每一对相邻区间的卡方值</p>
</li>
<li><p>将卡方值最小的一对区间合并</p>
<p>$$  \mathrm{X}^{2}=\sum_{i=1}^{2} \sum_{j=1}^{2} \frac{\left(A_{i j}-E_{i j}\right)^{2}}{E_{i j}}  $$</p>
<p>其中，Ai,j为第i区间第j类的实例的数量，Ei,j为期望频率（整体分布，整体分布计算的时候还是用的这俩区间的数值）=Ni*(Cj/N)，Cj为j类在整体分布中的比例，Ni为i区间样本数</p>
<p>例如label有0和1，有两个区间，即[[3,4], [5,7]]</p>
<p>A11=3,A12=4… E11=7*8/19</p>
</li>
</ul>
</li>
</ul>
</li>
<li><p>原理：假设这俩区间没差异，其卡方值代表了该假设（由实际分布得出的统计量）与理论（整体样本[特征]分布）的差异程度，越小则说明俩越接近，假设越成立。</p>
<p>评分卡模型中，第i区间第j类的实例：类是指区间中有多少类特征，实例指其对应的label</p>
</li>
<li><p>阈值：显著性水平+自由度（比类别数量少1）确定。</p>
</li>
</ul>

      
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              <div class="post-toc-content"><ol class="nav"><li class="nav-item nav-level-3"><a class="nav-link" href="#统计检验"><span class="nav-number">1.</span> <span class="nav-text">统计检验</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#T"><span class="nav-number">2.</span> <span class="nav-text">T</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#F"><span class="nav-number">3.</span> <span class="nav-text">F</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#卡方"><span class="nav-number">4.</span> <span class="nav-text">卡方</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#卡方检验"><span class="nav-number">4.1.</span> <span class="nav-text">卡方检验</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#卡方分布"><span class="nav-number">4.2.</span> <span class="nav-text">卡方分布</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#卡方分箱"><span class="nav-number">4.3.</span> <span class="nav-text">卡方分箱</span></a></li></ol></li></ol></div>
            

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  <script type="text/javascript">
    // Popup Window;
    var isfetched = false;
    var isXml = true;
    // Search DB path;
    var search_path = "search.xml";
    if (search_path.length === 0) {
      search_path = "search.xml";
    } else if (/json$/i.test(search_path)) {
      isXml = false;
    }
    var path = "/" + search_path;
    // monitor main search box;

    var onPopupClose = function (e) {
      $('.popup').hide();
      $('#local-search-input').val('');
      $('.search-result-list').remove();
      $('#no-result').remove();
      $(".local-search-pop-overlay").remove();
      $('body').css('overflow', '');
    }

    function proceedsearch() {
      $("body")
        .append('<div class="search-popup-overlay local-search-pop-overlay"></div>')
        .css('overflow', 'hidden');
      $('.search-popup-overlay').click(onPopupClose);
      $('.popup').toggle();
      var $localSearchInput = $('#local-search-input');
      $localSearchInput.attr("autocapitalize", "none");
      $localSearchInput.attr("autocorrect", "off");
      $localSearchInput.focus();
    }

    // search function;
    var searchFunc = function(path, search_id, content_id) {
      'use strict';

      // start loading animation
      $("body")
        .append('<div class="search-popup-overlay local-search-pop-overlay">' +
          '<div id="search-loading-icon">' +
          '<i class="fa fa-spinner fa-pulse fa-5x fa-fw"></i>' +
          '</div>' +
          '</div>')
        .css('overflow', 'hidden');
      $("#search-loading-icon").css('margin', '20% auto 0 auto').css('text-align', 'center');

      $.ajax({
        url: path,
        dataType: isXml ? "xml" : "json",
        async: true,
        success: function(res) {
          // get the contents from search data
          isfetched = true;
          $('.popup').detach().appendTo('.header-inner');
          var datas = isXml ? $("entry", res).map(function() {
            return {
              title: $("title", this).text(),
              content: $("content",this).text(),
              url: $("url" , this).text()
            };
          }).get() : res;
          var input = document.getElementById(search_id);
          var resultContent = document.getElementById(content_id);
          var inputEventFunction = function() {
            var searchText = input.value.trim().toLowerCase();
            var keywords = searchText.split(/[\s\-]+/);
            if (keywords.length > 1) {
              keywords.push(searchText);
            }
            var resultItems = [];
            if (searchText.length > 0) {
              // perform local searching
              datas.forEach(function(data) {
                var isMatch = false;
                var hitCount = 0;
                var searchTextCount = 0;
                var title = data.title.trim();
                var titleInLowerCase = title.toLowerCase();
                var content = data.content.trim().replace(/<[^>]+>/g,"");
                var contentInLowerCase = content.toLowerCase();
                var articleUrl = decodeURIComponent(data.url);
                var indexOfTitle = [];
                var indexOfContent = [];
                // only match articles with not empty titles
                if(title != '') {
                  keywords.forEach(function(keyword) {
                    function getIndexByWord(word, text, caseSensitive) {
                      var wordLen = word.length;
                      if (wordLen === 0) {
                        return [];
                      }
                      var startPosition = 0, position = [], index = [];
                      if (!caseSensitive) {
                        text = text.toLowerCase();
                        word = word.toLowerCase();
                      }
                      while ((position = text.indexOf(word, startPosition)) > -1) {
                        index.push({position: position, word: word});
                        startPosition = position + wordLen;
                      }
                      return index;
                    }

                    indexOfTitle = indexOfTitle.concat(getIndexByWord(keyword, titleInLowerCase, false));
                    indexOfContent = indexOfContent.concat(getIndexByWord(keyword, contentInLowerCase, false));
                  });
                  if (indexOfTitle.length > 0 || indexOfContent.length > 0) {
                    isMatch = true;
                    hitCount = indexOfTitle.length + indexOfContent.length;
                  }
                }

                // show search results

                if (isMatch) {
                  // sort index by position of keyword

                  [indexOfTitle, indexOfContent].forEach(function (index) {
                    index.sort(function (itemLeft, itemRight) {
                      if (itemRight.position !== itemLeft.position) {
                        return itemRight.position - itemLeft.position;
                      } else {
                        return itemLeft.word.length - itemRight.word.length;
                      }
                    });
                  });

                  // merge hits into slices

                  function mergeIntoSlice(text, start, end, index) {
                    var item = index[index.length - 1];
                    var position = item.position;
                    var word = item.word;
                    var hits = [];
                    var searchTextCountInSlice = 0;
                    while (position + word.length <= end && index.length != 0) {
                      if (word === searchText) {
                        searchTextCountInSlice++;
                      }
                      hits.push({position: position, length: word.length});
                      var wordEnd = position + word.length;

                      // move to next position of hit

                      index.pop();
                      while (index.length != 0) {
                        item = index[index.length - 1];
                        position = item.position;
                        word = item.word;
                        if (wordEnd > position) {
                          index.pop();
                        } else {
                          break;
                        }
                      }
                    }
                    searchTextCount += searchTextCountInSlice;
                    return {
                      hits: hits,
                      start: start,
                      end: end,
                      searchTextCount: searchTextCountInSlice
                    };
                  }

                  var slicesOfTitle = [];
                  if (indexOfTitle.length != 0) {
                    slicesOfTitle.push(mergeIntoSlice(title, 0, title.length, indexOfTitle));
                  }

                  var slicesOfContent = [];
                  while (indexOfContent.length != 0) {
                    var item = indexOfContent[indexOfContent.length - 1];
                    var position = item.position;
                    var word = item.word;
                    // cut out 100 characters
                    var start = position - 20;
                    var end = position + 80;
                    if(start < 0){
                      start = 0;
                    }
                    if (end < position + word.length) {
                      end = position + word.length;
                    }
                    if(end > content.length){
                      end = content.length;
                    }
                    slicesOfContent.push(mergeIntoSlice(content, start, end, indexOfContent));
                  }

                  // sort slices in content by search text's count and hits' count

                  slicesOfContent.sort(function (sliceLeft, sliceRight) {
                    if (sliceLeft.searchTextCount !== sliceRight.searchTextCount) {
                      return sliceRight.searchTextCount - sliceLeft.searchTextCount;
                    } else if (sliceLeft.hits.length !== sliceRight.hits.length) {
                      return sliceRight.hits.length - sliceLeft.hits.length;
                    } else {
                      return sliceLeft.start - sliceRight.start;
                    }
                  });

                  // select top N slices in content

                  var upperBound = parseInt('1');
                  if (upperBound >= 0) {
                    slicesOfContent = slicesOfContent.slice(0, upperBound);
                  }

                  // highlight title and content

                  function highlightKeyword(text, slice) {
                    var result = '';
                    var prevEnd = slice.start;
                    slice.hits.forEach(function (hit) {
                      result += text.substring(prevEnd, hit.position);
                      var end = hit.position + hit.length;
                      result += '<b class="search-keyword">' + text.substring(hit.position, end) + '</b>';
                      prevEnd = end;
                    });
                    result += text.substring(prevEnd, slice.end);
                    return result;
                  }

                  var resultItem = '';

                  if (slicesOfTitle.length != 0) {
                    resultItem += "<li><a href='" + articleUrl + "' class='search-result-title'>" + highlightKeyword(title, slicesOfTitle[0]) + "</a>";
                  } else {
                    resultItem += "<li><a href='" + articleUrl + "' class='search-result-title'>" + title + "</a>";
                  }

                  slicesOfContent.forEach(function (slice) {
                    resultItem += "<a href='" + articleUrl + "'>" +
                      "<p class=\"search-result\">" + highlightKeyword(content, slice) +
                      "...</p>" + "</a>";
                  });

                  resultItem += "</li>";
                  resultItems.push({
                    item: resultItem,
                    searchTextCount: searchTextCount,
                    hitCount: hitCount,
                    id: resultItems.length
                  });
                }
              })
            };
            if (keywords.length === 1 && keywords[0] === "") {
              resultContent.innerHTML = '<div id="no-result"><i class="fa fa-search fa-5x" /></div>'
            } else if (resultItems.length === 0) {
              resultContent.innerHTML = '<div id="no-result"><i class="fa fa-frown-o fa-5x" /></div>'
            } else {
              resultItems.sort(function (resultLeft, resultRight) {
                if (resultLeft.searchTextCount !== resultRight.searchTextCount) {
                  return resultRight.searchTextCount - resultLeft.searchTextCount;
                } else if (resultLeft.hitCount !== resultRight.hitCount) {
                  return resultRight.hitCount - resultLeft.hitCount;
                } else {
                  return resultRight.id - resultLeft.id;
                }
              });
              var searchResultList = '<ul class=\"search-result-list\">';
              resultItems.forEach(function (result) {
                searchResultList += result.item;
              })
              searchResultList += "</ul>";
              resultContent.innerHTML = searchResultList;
            }
          }

          if ('auto' === 'auto') {
            input.addEventListener('input', inputEventFunction);
          } else {
            $('.search-icon').click(inputEventFunction);
            input.addEventListener('keypress', function (event) {
              if (event.keyCode === 13) {
                inputEventFunction();
              }
            });
          }

          // remove loading animation
          $(".local-search-pop-overlay").remove();
          $('body').css('overflow', '');

          proceedsearch();
        }
      });
    }

    // handle and trigger popup window;
    $('.popup-trigger').click(function(e) {
      e.stopPropagation();
      if (isfetched === false) {
        searchFunc(path, 'local-search-input', 'local-search-result');
      } else {
        proceedsearch();
      };
    });

    $('.popup-btn-close').click(onPopupClose);
    $('.popup').click(function(e){
      e.stopPropagation();
    });
    $(document).on('keyup', function (event) {
      var shouldDismissSearchPopup = event.which === 27 &&
        $('.search-popup').is(':visible');
      if (shouldDismissSearchPopup) {
        onPopupClose();
      }
    });
  </script>





  

  

  

  

  

  

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